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Vocabulary
Compete the table.
Word  Definition 
Quadratic Inequality  ___________________________________________________ 
___________  the values of x that make y equal to zero in a quadratic function 
Polynomial Inequality:  ___________________________________________________ 
___________  A function which may be expressed as a ratio of two polynomials, specified to be greater or less than a given value 
Quadratic Inequalities
To solve quadratic inequalitites, you can graph it and see visually where the inequality is true. Without graphing, you can follow these steps:

Set up the inequality in the form (or
p(x)<0,p(x)≤0,p(x)≥0 ) 
Find the solutions to the equation
p(x)=0 . 
Divide the number line into intervals based on the solutions to
p(x)=0 .  Use test points to find solution sets to the equation.
Example
Find the solution set for the equation .
First, factor:
Then create the intervals:
Finally, make a chart.
Interval  Test Point 
Is 
Part of Solution set? 













if and only if
.
For more on quadratic inequalities, click here.
Polynomial and Rational Inequalities
In polynomial and rational inequalities, the four basic steps are the same as in polynomial inequalities. For rational inequalities, however, you must also account for possible sign changes at vertical asymptotes or a break in the graph.
Add the vertical asymtotes to your intervals, and create the same chart as you did for quadratic inequalities.
.
For more on polynomial and rational inequalities, click here.
Practice
1) Find the solution set of the inequality
2) Find the solution set:
3) Find the solution set:
4) Graph the solution set:
.
Find the solution set of the following inequalities without using a calculator. Display the solution set on the number line.

x2+2x−3≤0 
−6x2−13x+5≥0 
1−xx<1 
x44−x2<0 
4x3−8x2−x+2≥0
Solve the following inequalities:

n3−2n2−n+2n3+3n2+4n+12<0 
n3+3n2−4n−12n3−5n2+4n−20≤0 
2n3+5n2−18n−453n3−n2+27n−9≥0